Discrete Fourier Transform (DFT) processors are one the most important class of specialized computing engines used for signal and image processing in disciplines ranging from electronic warfare (EW), cyber-security, optical and wireless communications, image processing and spectroscopy. Conventional DFT processor implementation rests on reduced-complexity algorithms such as split-radix FFT (SRFFT) that approach the lower bound on number of multiplications predicted by Winograd (Mathematics of Computation, 32(141):175-199, 1978). However, a typical 256-bit FFT operation still uses approximately 2000 real multiples and more than 4000 additions (H. Sorensen et al., IEEE Transactions on Acoustics, Speech and Signal Processing, Vol. 34, pp. 152 1986). The multiplier operation carries the largest energy cost in modern digital signal processor (DSP) architecture: it is nearly 300% more expensive than the addition and more than 10 times than a Boolean operation. The energy scaling supported by Moore's law in the past is no longer applicable—while multiplier cost was 1.5 pJ in fast (10-TIOPS) 32 nm-CMOS, the new (12 nm) technology draws only marginally less power (S. Savory, “Digital Signal Processing for Coherent Systems,” OFC/NFOEC Technical Digest, OTh3C7, 2012). CMOS feature scaling has finally met the gate leakage barrier that prevents, even in principle, achieving low dissipation at high data processing speeds. Indeed, the CMOS gate has become so small that it cannot be effectively switched off. As a consequence, energy-intensive processors (such as multiplier-dominated DFT) are now facing strict speed limit since generated heat cannot be dissipated in practical manner.
In an alternative approach, a Fourier optical processor based on free-space optics (defined as an architecture composed of 3- or 2-dimensional lenses, optical grating and combining elements) can calculate spatial or temporal transform. However, five decades since its initial introduction (J. W. Goodman, Introduction to Fourier Optics, McGraw-Hill (1968)), no practical computing architecture using optical co-processor has been implemented or used in commercial systems. Reasons behind this apparent failure are both fundamental and practical; we only point to widely recognized speed and size limits that free-space optics imposes on any architecture. Indeed, even when combined with electrooptical or optomechanical primitives, free-space Fourier systems are large, dissipative and slow in comparison to its modern electronics counterparts.
In one important DFT application, signal detection, classification and interpretation (DCI) across the entire radio-frequency (RF) spectrum poses both fundamental and technological challenges. The advent of fast RF modulation has enabled a diversified family of ultrawideband (UWB) transceivers. The inherent advantages of UWB link include power-efficient transmission, resistance to interference, multipath distortion, and band-reuse. A common feature in all systems, defined by regulations, is large physical bandwidth, exceeding 0.5 GHz in all cases. In practice, this means that a sub-nanosecond pulse is generated and manipulated in order to code channel either in time or frequency domain.
An early UWB approach focused on time-modulated (TM) UWB emitters, resembling, in its simplest form, well-known pulse-position modulation (PPM). In contrast to classical PPM scheme, often used with deep-space optical links, TM-UWB time reference is often dithered by a tailored quasi-random sequence in order to suppress channel spectral density and equalize it across the allocated band. Similarly, a frequency-hopping (FH) coding was also developed, particularly to minimize spectral channel density at “crowded” bands and to provide for adaptive spectral provisioning. In both schemes, strict temporal reference is absent and the correlator plays a critical role within the receiver. An incoherent nature of such UWB link also puts high spectral efficiency coding such as pulse-amplitude (PAM) modulation at distinct disadvantage.
In addition to complying to non-interference regulation, useful wireless transceiver technologies must also bypass multipath propagation, interference from other devices, intentional jamming, and provide either physical or coding security layer. Consequently, conventional (time-referenced) PPM, formed with ultrashort pulses, cannot achieve these requirements. However, if UWB PPM is combined with fast frequency hopping, then goals of regulation-compliant spectral utilization, multi-access links, and low probability of intercept become realistic. The combination of frequency hopping (FH) with PPM could not only eliminate fixed harmonics seen in fixed time-frame TM-UWB, but could also add a level of link security not inherent in adaptive-band UWB links.
The qualities that make FH-PPM modulation attractive also pose a significant reception problem. Firstly, UWB pulses with bandwidth in excess of GHz need to be utilized in order to spread the signal in frequency domain and suppress the spectral power density below the FCC-specified power threshold. Secondly, an introduction of frequency-hopping mandates additional bandwidth at the receiver, in excess of PPM-only detection. While the excess frequency bandwidth requirement can be addressed by incorporating a set of local oscillator's and mixers at the receiver, this solution is not applicable in dissipation-constrained receiver architectures, and particularly those in satellite and remoted devices. Thus, to address the bandwidth challenge, an analog to digital converter (ADC) possessing sufficient speed must be utilized. While possible, at least in principle, the use of high-bandwidth ADC imposes practical challenges for multi-GHz-wide channels. Conversely, when physical bandwidth of FH-PPM link exceeds 10 GHz, this challenge also becomes fundamental as it induces runaway dissipation requirement.
Firstly, high noise figure would be induced due to the need for high-count signal splitting and subsequent amplification. Secondly, precise filter alignment would have to be realized regardless of how fine frequency pitch is required. The latter is in direct conflict with the need for high filter-to-filter isolation necessary to suppress the crosstalk. In contrast, photonic-assisted front-ends have been demonstrated in the past, overcome both the performance and implementation limits. Specifically, the use of modulated optical frequency combs and parametric signal multicasters have been widely studied and have addressed high insertion loss, distortion and frequency programmability challenges.
Pulse position modulation (PPM) has been used in radio-frequency domain to achieve both low-dissipation requirements and provide precision ranging. In ultra-wide band (UWB) architectures, it underpins asynchronous receiver, multiple access environments and interference-resistant transmission. When combined with frequency hopping (FH), it allows for additional level of immunity to jamming and low probability of intercept. Realization of frequency-hopping PPM (FH-PPM) transceiver poses practical challenge, particularly in UWB RF range. With UWB pulses reaching the multi-GHz range, frequency hopping adds to the effective bandwidth at which receiver must be operated, exceeding the performance of modern quantizer and digital demodulation backplane.
Another approach for addressing UWB challenges involves cyclostationary analysis, which lies at the core of EW and signal intelligence (SIGINT) intercept systems (W. A. Gardner, et al., “Cyclostationarity: Half a century of research,” Signal Proc., vol. 86, no. 4, pp. 639-697, April 2006). Introduced nearly four decades ago, cyclostationary analysis can intercept and classify a modulated waveform from a background signal such as noise or jamming. To accomplish this, spectral computation must be performed over multiple modulation cycles. The received signal is first digitized using an analog-to-digital converter (ADC) and then subsequently mapped to the Fourier domain (FFT). After the spectral representation is obtained, spectral correlation is computed in order to generate the two-dimensional spectral correlation function (SCF) representation, discriminating the noise. A wideband ADC poses the first processing challenge that can be quantified in terms of precision, operating bandwidth and dissipation. While an ADC capable of contiguous RF range is unlikely to be constructed anytime soon, circuits operating beyond 20 GHz have been reported. If we assume that an RF bandwidth of 100 GHz can be addressed by a combination of multiple ADC stages, such a compound digitizer would still dissipate nearly 100 Watts. Even if this were acceptable in a select set of CS applications, the effective number of bits of such a digitizer would strictly limit its utility. Thus, in practice, current, all-electronics DFT technology limits real-time spectral bandwidth to sub-GHz-scale range—two orders of magnitude below the needs of future EW spectral range (>110 GHz).
Yet another application of DFTs is real-time pattern recognition, which has encountered technology limits in imaging, SIGINT and cyber-defense fields. While image-based pattern recognition is well-known, a less-well known, but equally important challenge is posed by Terabit-per-second-capacity lightwave channel that carries high-capacity terrestrial and submarine fiber data traffic. To analyze an anomalous traffic pattern (such as denial of service or network domain scanning) in such high-capacity fiber link in real time, it is necessary to recognize and intercept a specific bit sequence on the fly (at time scale comparable to the flight of the lightwave packet over few kilometers of fiber). In practical terms, this means that DFT of Terabit-class stream must be computed at “wirespeed”, i.e., at a latency that is comparable to a lightwave packet traversing the localized segment of fiber. Currently, any DFT of data traffic is performed off-line (store-and-compute), thus precluding, even in principle, real-time network traffic analysis. A DFT coprocessor capable of real-time, continuous operation that matches a lightwave channel rate would dramatically change the very nature of modern cyber defense strategy
The above-described applications provide just a few examples of the significant limitations that electronic- and computation-based approaches to DFT and other transforms have imposed on a wide range of technologies. Accordingly, the need remains for an approach to reduce this computational bottleneck to allow real-time, wideband execution of signal processing algorithms.